Si,j = 2 if the residue at position i of sequence #1
is the same as the residue at position j of sequence #2 (match score); otherwise

Si,j = -1 (mismatch score)

w = -2 (gap penalty)

Initialization Step

The first step in the global alignment dynamic programming approach is to
create a matrix with M + 1 columns and N + 1 rows where M and N correspond to
the size of the sequences to be aligned.

The first row and first column of the matrix can be initially filled with 0.

Matrix Fill Step

One possible (inefficient) solution of the matrix fill step finds the
maximum global alignment score by starting in the upper left hand corner in
the matrix and finding the maximal score Mi,j for each position in the
matrix. In order to find Mi,j for any i,j it is minimal to know the
score for the matrix positions to the left, above and diagonal to i, j. In
terms of matrix positions, it is necessary to know Mi-1,j,
Mi,j-1 and Mi-1, j-1.

For each position, Mi,j is defined to be the maximum score
at position i,j; i.e.

Note that in the example, Mi-1,j-1 will be red,
Mi,j-1 will be green and Mi-1,j will be blue.

Using this information, the score at position 1,1 in the matrix can be
calculated. Since the first residue in both sequences is a G, S1,1
= 2, and by the assumptions stated earlier, w = -2. Thus,
M1,1 = MAX[M0,0 + 2, M1,0 - 2, M0,1
- 2] = MAX[2, -2, -2].

A value of 2 is then placed in position 1,1 of the scoring matrix.
Note that there is also an arrow placed back into the cell that resulted
in the maximum score, M[0,0].

Moving down the first column to row 2, we can see that there is once again
a match in both sequences. Thus, S1,2 = 2. So M1,2 =
MAX[M0,1 + 2, M1,1 - 2, M0,2 -2] =
MAX[0 + 2, 2 - 2, 0 - 2] = MAX[2, 0, -2].

A value of 2 is then placed in position 1,2 of the scoring matrix and an
arrow is placed to point back to M[0,1] which led to the maximum score.

Note that in the above case, there are two different ways to get the
maximum score. In such a case, pointers are placed back to all of the cells
that can produce the maximum score.

The rest of the score matrix can then be filled in. The completed score
matrix will be as follows:

Traceback Step

After the matrix fill step, the maximum global alignment score for the two
sequences is 3. The traceback step will determine the actual alignment(s)
that result in the maximum score.

The traceback step begins in the M,J position in the matrix, i.e. the position
where both sequences are globally aligned.

Since we have kept pointers back to all possible predacessors, the traceback
step is simple. At each cell, we look to see where we move next according to
the pointers. To begin, the only possible predacessor is the diagonal match.

This gives us an alignment of

A
|
A

Note that the blue letters and gold arrows indicate the path leading to the
maximum score.

We can continue to follow the path using a single pointer until we get to the
following situation.

The alignment at this point is

T C A G T T A
| | | |
T C _ G _ _ A

Note that there are now two possible neighbors that could result in the
current score. In such a case, one of the neighbors is arbitrarily
chosen.

Once the traceback is completed, it can be seen that there are only two
possible paths leading to a maximal global alignment.

One possible path is as follows:

This gives an alignment of

G A A T T C A G T T A
| | | | | |
G G A _ T C _ G _ _ A

The other possible path is as follows:

This gives an alignment of

G A A T T C A G T T A
| | | | | |
G G A T _ C _ G _ _ A

Remembering that the scoring scheme is +2 for a match, -1 for a mismatch,
and -2 for a gap, both sequences can be tested to make sure that they
result in a score of 3.